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Large Liquefied Natural Gas (LNG) tanks are prone to damage during strong earthquakes, and accurate seismic analysis must be performed during the design phase to prevent secondary disasters. However, the seismic analysis of large LNG tanks is associated with high computational requirements, which cannot be satisfied by the calculation efficiency of traditional analytical techniques such as the Coupled Eulerian–Lagrangian (CEL) method. Thus, this paper aims to employ a less computationally demanding algorithm, the Smoothed Particle Hydrodynamics-Finite Element Method (SPH-FEM) algorithm, to simulate large LNG tanks. The seismic response of a 160,000 m^{3} LNG prestressed storage tank is evaluated with different liquid depths using the SPH-FEM algorithm, and simulation results are obtained with excellent efficiency and accuracy. In addition, large von Mises stress at the base of the tank indicates that strong earthquakes can severely jeopardize the structural integrity of large LNG tanks. Therefore, the SPH-FEM algorithm provides a feasible approach for the analysis of large liquid tanks in seismic engineering applications.

Natural gas is a reliable source of energy that is used globally to meet growing energy demands. With the increasing consumption of natural gas over the past few years, Liquefied Natural Gas (LNG) tanks have become a major component of urban infrastructure. As a result, the scale of tank construction has also increased, but this change has been associated with various safety risks. LNG storage tanks have high seismic risks compared to traditional buildings because they can lead to secondary disasters, such as explosions and environmental pollution, that result in significant property damage or loss of life. In addition, LNG tanks are more vulnerable to earthquakes owing to their low redundancy, low ductility, and low energy-dissipating capacity compared to conventional structures [

To analyze the dynamic performance of large LNG tanks, it is essential to understand fluid–structure interaction (FSI) and dynamic performance of the structure. Different from conventional structures, seismic analysis of liquid storage tanks must account for fluid–structure interaction as a result of inertial earthquake loading and hydrodynamic pressure. The seismic design of storage tanks was initially established on the rigid wall model proposed by Housner [

Based on applicable theories regarding fluid–structure interaction, since the 1950s, there have been several studies on the dynamic analysis of LNG storage tanks. Hwang [^{3} LNG prestressed concrete outer tank under impact loading, and types of impact damage were determined based on dynamic response results including stress, displacement, energy, and critical impact velocity. Li et al. [

The aforementioned numerical simulation methods are very computationally intensive due to the sheer size of large LNG tanks. Given the recent rise in large LNG tank construction, it is essential to employ a fast analysis method that can simulate the seismic response of LNG tanks in practical engineering applications. Thus, an efficient smoothed particle hydrodynamics-finite element method (SPH-FEM) algorithm [

SPH is suitable for modeling fluid behaviour because it is a mesh-free method that discretizes a liquid field into particles [

Coupled with a FEM method for the solid component, an SPH-FEM algorithm greatly improves the calculation accuracy and efficiency for fluid–structure impact problems. The coupled SPH-FEM method was first proposed by Attaway et al. [

^{3} LNG storage tank under three earthquakes with the same site conditions.

The formulation of the SPH method is often divided into two parts: the integral representation of the field function and the particle approximation [

If a smoothing function

The normalization condition states that

The Delta function property is observed when the smoothing length approaches zero

The compact condition implies that

The basic governing fluid dynamics equations are based on three fundamental laws of conservation: conservation of mass, momentum, and energy. The governing equations for dynamic fluid flows can be written as a set of partial differential equations using the Lagrangian method; this system of partial differential equations is the famous Navier–Stokes equations, the governing equations of which are given as follows (see equations (

The continuity equation:

The momentum equation:

The energy equation:

A continuous density particle approximation method is employed to approximate the density. This approximation method is obtained by applying the SPH approximation concept to transform the continuity equation. Different forms of density approximation equations can be obtained by applying different conversions and operations to (

The particle approximation method for the momentum equation is similar to the abovementioned continuous density method, and some transformations are required. The momentum equation approximation can be derived in differential form based on different transformations. Equation (

There are several approximation expressions of the work performed by pressure; thus, the internal energy calculation for the work performed by pressure has many alternative forms. The following form (

Time stepping in ABAQUS is explicit and is limited by the Courant Condition as shown below (equation (^{−6} s, and the convergence of the simulation is guaranteed [

The main concept in SPH-FEM coupling is centered around the relationship between the two algorithms. In each time step, the velocity and displacement of FEM nodes are transferred to particles. Simultaneously, the coordinate and velocity of particles are transferred to FEM nodes. While particles supply boundary conditions for FEM, FEM maintains the continuity of the particles by preventing the boundary effect [

Computation flowchart of the SPH-FEM coupling method.

Compared to the SPH-FEM method, the coupled Eulerian–Lagrangian (CEL) method developed by Du [

A cubic water tank is taken as a model and is designed according to the following geometric specifications: the side length is 1 m, the wall thickness is 10 mm, and the liquid height is 0.8 m. The tank model is composed of concrete with the following material properties: density ^{9} N/m^{2}, and Poisson’s ratio _{m} is the internal energy per unit mass,

Material properties of water.

Material | Density (kg/m^{3}) | Sonic speed (m/s) | Dynamic viscosity (kg/(m·s)) | Coefficients of the equation of state | |
---|---|---|---|---|---|

Water | 1000 | 1480 | 0.001 | 0 | 0 |

The equation of state of

The water-filled models constructed based on the two algorithms are shown in Figures

Model for the SPH-FEM algorithm.

Model for the CEL method.

The results of the two algorithms during one period are compared based on three factors: sloshing wave pattern, stress distribution on container wall, and computational efficiency. Both methods can accurately reflect the sloshing wave pattern (as shown in Figure

Comparison of the sloshing pattern between the two algorithms. (a) At 0.8 s with the SPH-FEM algorithm, (b) at 0.8 s with the CEL method, (c) at 1.6 s with the SPH-FEM algorithm, and (d) at 1.6 s with the CEL method.

Using the two algorithms, the von Mises stress distribution on the container wall at 0.8 s and 1.6 s in one period is shown in Figure

Comparison of the von Mises stress on the walls between the two algorithms. (a) At 0.8 s with the SPH-FEM algorithm, (b) at 0.8 s with the CEL method, (c) at 1.6 s with the SPH-FEM algorithm, and (d) at 1.6 s with the CEL method.

As shown in Table

Comparison of the computational efficiencies of the two algorithms.

Analysis time (s) | Algorithm | Simulation time | Stable time step (s) |
---|---|---|---|

10 | SPH-FEM | 4 h 52 min 13 s | 0.0165 |

CEL | 8 h 58 min 14 s | 0.0274 |

This study employed a 160,000 m^{3} LNG prestressed storage tank, which was adapted from an LNG technical manual as shown in Figure

LNG storage tank cross section (unit: mm).

The inner tank has a total of 10 layers as shown in Figure

Thickness of each layer in the inner tank.

Layer | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Thickness (mm) | 24.9 | 22.4 | 19.8 | 17.3 | 14.7 | 12.2 | 12 | 12 | 12 | 12 |

The LNG storage tank consists of four parts: LNG, an inner tank, an insulation layer, and an outer tank. The inner tank is composed of 9% Ni steel. The tank exhibits excellent low-temperature resistance [

Material characteristics of the LNG prestressed storage tank.

Parameter/material | Density (kg/m^{3}) | Modulus of elasticity (MPa) | Poisson’s ratio | Yield strength (MPa) |
---|---|---|---|---|

Prestressed concrete | 2500 | 3.6 × 10^{10} | 0.2 | — |

Expanded perlite | 56 | 11.25 | 0.15 | — |

Resilient felt | 300 | 800 | 0.12 | — |

Rebar HRB400 | 7800 | 2 × 10^{11} | 0.3 | 400 |

Steel cable | 7800 | 1.95 × 10^{11} | 0.3 | 1860 |

9% Ni steel | 7850 | 2.06 × 10^{11} | 0.3 | 600 |

Material properties of LNG.

Material | Density (kg/m^{3}) | Sonic speed (m/s) | Dynamic viscosity (kg/(m·s)) | Coefficients of the equation of state | |
---|---|---|---|---|---|

г_{0} | |||||

LNG (liquid) | 480 | 1500 | 0.00113 | 0 | 0 |

ABAQUS software [

FEM of the LNG storage tank. (a) The outer tank. (b) The inner tank.

19T15S steel cables with a tensile strength of 1860 MPa are embedded in the protective cover of the tank. A total of 122 steel cables are vertically anchored at the bottom and top of the concrete wall, and 220 circumferential steel cables are anchored in separate half-circles on four vertical supporting columns arranged at 90° [

The basic principles of applying prestress in ABAQUS [

The bottom boundary of the model is fixed, and the interaction between soil and structure is not considered. The static condition of the LNG tank is primarily calculated based on self-weight to ensure that the LNG liquid reaches steady state. Natural frequencies are important parameters in the design of LNG prestressed storage tanks. There are three primary ways to extract eigenvalues in ABAQUS: the AMS eigensolver, the subspace iteration method, and the block Lanczos algorithm. Here, the first five frequencies of the empty tank are calculated with the Lanczos algorithm as 3.753 Hz, 4.902 Hz, 5.088 Hz, 5.097 Hz, and 5.204 Hz.

In China, seismic codes for large LNG storage tanks are based on the seismic codes of building structures [

Details of the 3 ground motions employed in this study.

Event | Station | Time |
---|---|---|

Imperial Valley-01 | El Centro Array #9 | 1940.5.18 |

Northridge-01 | Northridge-Saticoy | 1994.1.17 |

Kern County, California | Taft Lincoln School Tunnel | 1994.1.17 |

Acceleration response spectra of selected ground motions and of code specifications.

The entire simulation using the SPH-FEM algorithm is conducted using an ordinary personal computer. The processor is an Intel® Core™ i7-8700 CPU @ 4.10 GHz, and the installation memory is 16.00 GB. The von Mises stress distribution of the inner tank alone under hydrostatic conditions at different LNG liquid levels, without the effect of insulation layer and outer tank, is shown in Figure

Distribution of von Mises stress distribution on the inner tank under static conditions at different LNG liquid levels. (a) At 25% full, (b) at 50% full, (c) at 75% full, and (d) at 100% full.

Among the three seismic waves, Taft produces the greatest von Mises stress on the outer and inner tank at different LNG liquid levels. Therefore, Taft is chosen as the critical wave for this analysis. The stress distribution on the outer tank exhibits a symmetrical trend about the

The maximum von Mises stress distribution of the outer tank at different LNG liquid levels. (a) At 25% full, (b) at 50% full, (c) at 75% full, and (d) at 100% full.

The respective maximum stress of the inner tank with LNG liquid levels of 25%, 50%, 75%, and 100% is shown in Figure

The maximum von Mises stress distribution of the inner tank under different LNG liquid levels. (a) At 25% full, (b) at 50% full, (c) at 75% full, and (d) at 100% full.

According to postprocessing results, three representative paths, as shown in Figure

Three representative paths selected to analyze the von Mises stress distribution along the tank wall: A, B, and C.

Distribution of the von Mises stress on the outer tank with height along different paths. (a) Path A. (b) Path B. (c) Path C.

Distribution of von Mises stress on inner tank with respect to height along different paths. (a) Path A. (b) Path B. (c) Path C.

In addition, stress values along different paths are different for a given liquid depth, but the overall trends are the same. When the liquid level is less than 50%, the stress on the outer tank first increases and then decreases with an increase in height. When the liquid level exceeds 50%, the stress on the outer tank initially decreases, then increases, and decreases again. Moreover, stress in the lower portion of the tank is much larger than stress in the upper portion when the liquid level is greater than 50%. In addition, a greater LNG liquid volume leads to greater stored energy and greater stress at the base of the tank as shown in Figures

When comparing outer and inner tanks, it is observed that different liquid amounts can produce similar stress levels at a height of approximately 10 m. For the outer tank, the height that correspond to this stress level gradually decreases from Path A to Path C. In contrast, for the inner tank, the height that corresponds to this stress level is nearly identical across all paths.

The displacements of the highest point on the outer wall of the tank filled with 25%, 50%, 75%, and 100% LNG are shown in Figure

Tip displacement of the outer wall under different liquid volume. (a) El Centro, (b) Northridge, and (c) Taft.

Figure

Base shear of the tank under different liquid volume. (a) El Centro, (b) Northridge, and (c) Taft.

To verify the SPH-FEM simulation results under earthquake loading, they are compared with CEL simulation results in terms of tip displacements and base shear. The maximum tip displacement and the maximum base shear using the SPH-FEM and CEL method under El Centro, Taft, and Northridge seismic waves are shown in Figures

Comparison of the maximum tip displacement of the outer wall under different liquid volume between the two algorithms. (a) El Centro, (b) Northridge, and (c) Taft.

Comparison of the maximum base shear under different liquid volume between the two algorithms. (a) El Centro, (b) Northridge, and (c) Taft.

In this study, the SPH-FEM method is compared with the CEL method to demonstrate the advantage of SPH-FEM in terms of efficiency and accuracy. Then, the SPH-FEM algorithm is used to analyze the dynamic response of a 160,000 m^{3} LNG prestressed storage tank subjected to three earthquake waves in a Class II site. The main conclusions drawn from the numerical simulations can be summarized as follows:

Under static conditions, the von Mises stress increases at a linear rate with an increasing liquid volume. Under dynamic conditions, the von Mises stress increases at a nonlinear rate.

The stresses produced on the outer and inner tanks under earthquake loading are very similar at a height of approximately 10 m for each liquid volume. The maximum tip displacement of the outer tank is 4.16 mm, and the maximum base shear is

The maximum stress of the inner tank with 100% LNG liquid level exceeds 500 MPa under the chosen seismic waves and raises concerns for its structural safety. Development of an improved structural system and a warning mechanism is of paramount importance to the design of LNG tanks at high liquid levels.

The SPH-FEM algorithm is accurate and more efficient than the CEL method. Moreover, the SPH-FEM algorithm exhibits excellent capability for simulating large storage tanks under a reasonable time to provide a reference for the design and construction of large LNG tanks.

The largest LNG prestressed storage tank in China is currently 200,000 m^{3}. The sheer size of the tank requires a significant amount of meshing during seismic simulation, which is almost impossible to accomplish on an ordinary computer using a mesh-based method. The SPH-FEM algorithm provides a feasible method to simulate extremely large LNG tanks on a personal computer, which can help designers obtain its seismic performance in an acceptable time range without significant reliance on hardware capabilities. Ongoing work is being performed to optimize the design of large LNG tanks under different earthquakes using the SPH-FEM algorithm and will be reported at a later time.

Domain of integration

Kernel function

Smoothing length

Density, kg/ m^{3}

Time step size

Number of particles

Gravitational acceleration

Mass of particle

_{s}:

Local speed of sound

_{m}:

Internal energy per unit mass

Volumetric compressibility

Stress

Velocity

Direction of coordinates

Direction of coordinates

Poisson’s ratio

Small positive number

The

The

The data of ground motions used to support the findings of this study have been deposited in the PEER Ground Motion Database.

Any opinions, findings, conclusions, and recommendations expressed here are those of the authors and do not necessarily reflect the views of the sponsors.

The authors declare that they have no conflicts of interest regarding the publication of this paper.

This research was supported by the China Scholarship Council, the Natural Science Foundation of China (Grant no. 51738007), and the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery (201606290) Program. The authors would like express their appreciation to Professor Ozden Turan and Professor Eren Semercigil for their input.

^{3}full capacity LNG storage tank in China